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Pfaffian Schur Processes and Last Passage Percolation in a Half-Quadrant1 The Annals of Probability 2018, Vol. 46, No. 6, 3015–3089 https://doi.org/10.1214/17-AOP1226 © Institute of Mathematical Statistics, 2018 PFAFFIAN SCHUR PROCESSES AND LAST PASSAGE PERCOLATION IN A HALF-QUADRANT1 ∗ BY JINHO BAIK ,2,GUILLAUME BARRAQUAND†,3,IVA N CORWIN†,4 AND TOUFIC SUIDAN University of Michigan∗ and Columbia University† We study last passage percolation in a half-quadrant, which we analyze within the framework of Pfaffian Schur processes. For the model with expo- nential weights, we prove that the fluctuations of the last passage time to a point on the diagonal are either GSE Tracy–Widom distributed, GOE Tracy– Widom distributed or Gaussian, depending on the size of weights along the diagonal. Away from the diagonal, the fluctuations of passage times fol- low the GUE Tracy–Widom distribution. We also obtain a two-dimensional crossover between the GUE, GOE and GSE distribution by studying the mul- tipoint distribution of last passage times close to the diagonal when the size of the diagonal weights is simultaneously scaled close to the critical point. We expect that this crossover arises universally in KPZ growth models in half-space. Along the way, we introduce a method to deal with diverging cor- relation kernels of point processes where points collide in the scaling limit. CONTENTS 1. Introduction ............................................3016 1.1. Previous results in (half-space) LPP ............................3017 1.2. Previous methods ......................................3018 1.3. Our methods and new results ................................3018 1.4.OthermodelsrelatedtoKPZgrowthinahalf-space...................3020 1.5.Mainresults.........................................3020 Outline of the paper ........................................3024 2. Fredholm Pfaffian formulas ...................................3024 2.1.GUETracy–Widomdistribution..............................3024 2.2. Fredholm Pfaffian ......................................3025 2.3.GOETracy–Widomdistribution..............................3026 2.4.GSETracy–Widomdistribution..............................3027 Received July 2016; revised August 2017. 1Supported by NSF PHY11-25915. 2Supported in part by NSF Grant DMS-1361782. 3Supported in part by the LPMA UMR CNRS 7599, Université Paris-Diderot–Paris 7 and the Packard Foundation through I. Corwin’s Packard Fellowship. 4Supported in part by the NSF through DMS-1208998, a Clay Research Fellowship, the Poincaré Chair and a Packard Fellowship for Science and Engineering. MSC2010 subject classifications. Primary 60K35, 82C23; secondary 60G55, 05E05, 60B20. Key words and phrases. Last passage percolation, KPZ universality class, Tracy–Widom distri- butions, Schur process, Fredholm Pfaffian, phase transition. 3015 3016 BAIK, BARRAQUAND, CORWIN AND SUIDAN 2.5.Crossoverkernels......................................3029 3.PfaffianSchurprocess......................................3031 3.1. Definition of the Pfaffian Schur process ..........................3031 3.1.1. Partitions and Schur positive specializations ....................3031 3.1.2. Useful identities ...................................3033 3.1.3. Definition of the Pfaffian Schur process ......................3033 3.2.PfaffianSchurprocessesindexedbyzig-zagpaths....................3036 3.3.DynamicsonPfaffianSchurprocesses...........................3037 3.4.Thefirstcoordinatemarginal:Lastpassagepercolation.................3041 4. Fredholm Pfaffian formulas for k-pointdistributions......................3044 4.1.Pfaffianpointprocesses...................................3045 4.2.Half-spacegeometricweightLPP.............................3046 4.3.Deformationofcontours..................................3048 4.4. Half-space exponential weight LPP ............................3050 5.ConvergencetotheGSETracy–Widomdistribution......................3058 5.1.AformalrepresentationoftheGSEdistribution.....................3059 5.2. Asymptotic analysis of the kernel Kexp ..........................3068 5.3.Conclusion..........................................3073 6.Asymptoticanalysisinotherregimes:GUE,GOE,crossovers................3074 6.1. Phase transition .......................................3074 6.2.GOEasymptotics......................................3076 6.3.GUEasymptotics......................................3078 6.4.Gaussianasymptotics....................................3084 6.5. Convergence of the symplectic-unitary transition to the GSE distribution .......3086 Acknowledgments..........................................3087 References..............................................3087 1. Introduction. In this paper, we study last passage percolation in a half- quadrant of Z2. We extend all known results for the case of geometric weights (see discussion of previous results below) to the case of exponential weights. We study in a unified framework the distribution of passage times on and off diagonal, and for arbitrary boundary condition. We do so by realizing the joint distribution of last passage percolation times as a marginal of Pfaffian Schur processes. This allows us to use powerful methods of Pfaffian point processes to prove limit theorems. Along the way, we discuss an issue arising when a simple point process converges to a point process where every point has multiplicity two, which we expect to have independent interest. These results also have consequences about interacting particle systems—in particular the TASEP on positive integers and the facilitated TASEP—that we discuss in [2]. DEFINITION 1.1 (Half-space exponential weight LPP). Let (wn,m)n≥m≥0 be a sequence of independent exponential random variables5 with rate 1 when n ≥ m + 5The exponential distribution with rate α ∈ (0, +∞), denoted E(α), is the probability distribution −αx on R>0 such that if X ∼ E(α), P(X > x) = e . PFAFFIAN SCHUR PROCESSES AND LPP IN A HALF-QUADRANT 3017 FIG.1. LPP on the half-quadrant. One admissible path from (1, 1) to (n, m) is shown in dark gray. H(n,m) is the maximum over such paths of the sum of the weights wij along the path. 1 and with rate α when n = m. We define the exponential last passage percolation (LPP) time on the half-quadrant, denoted H(n,m), by the recurrence for n ≥ m, max H(n− 1,m),H(n,m− 1) if n ≥ m + 1, H(n,m)= wn,m + H(n,m− 1) if n = m with the boundary condition H(n,0) = 0. It is useful to notice that the model is equivalent to a model of last passage percolation on the full quadrant where the weights are symmetric with respect to the diagonal (wij = wji). REMARK 1.2. If one imagines that the light gray area in Figure 1 corresponds to the percolation cluster at some time, its border (shown in black in Figure 1) evolves as the height function in the Totally Asymmetric Simple Exclusion process on the positive integers with open boundary condition, so that all our results could be equivalently phrased in terms of the latter model. 1.1. Previous results in (half-space) LPP. The history of fluctuation results in last passage percolation goes back to the solution to Ulam’s problem about the asymptotic distribution of the size of the longest increasing subsequence in a uni- formly random permutation [4]. A proof of the nature of fluctuations in exponential distribution LPP is provided in [29]. On a half-space, [5–7] studies the longest in- creasing subsequence in random involutions. This is equivalent to a half-space LPP problem since for an involution σ ∈ Sn, the graph of i → σ(i) is symmetric with respect to the first diagonal. Actually, [5–7] treat both the longest increasing sub- sequences of a random involution and symmetrized LPP with geometric weights. That work contains the geometric weight half-space LPP analogue of Theorem 1.3. 3018 BAIK, BARRAQUAND, CORWIN AND SUIDAN The methods used therein only worked when restricted to the one point distribu- tion of passage times exactly along the diagonal. Further work on half-space LPP was undertaken in [38], where the results are stated there in terms of the discrete polynuclear growth (PNG) model rather than the equivalent half-space LPP with geometric weights. The framework of [38] allows to study the correlation kernel corresponding to multiple points in the space direction, with or without nucleations at the origin. In the large scale limit, the authors performed a nonrigorous critical point derivation of the limiting correlation kernel in certain regimes and found Airy-type process and various limiting crossover kernels near the origin, much in the same spirit as our results (see the discussion about these results in Section 1.3). 1.2. Previous methods. A key property in the solution of Ulam’s problem in [4] was the relation to the RSK algorithm which reveals a beautiful algebraic struc- ture that can be generalized using the formalism of Schur measures [34] and Schur processes [35]. RSK is just one of a variety of Markov dynamics on sequences of partitions which preserves the Schur process [13, 17]. Studying these dynamics gives rise to a number of interesting probabilistic models related to Schur pro- cesses. In the early works of [4, 6], the convergence to the Tracy–Widom distribu- tions was proved by Riemann–Hilbert problem asymptotic analysis. Subsequently, Fredholm determinants became the preferred vehicle for asymptotic analysis. In a half-space, [5] applied RSK to symmetric matrices. Subsequently, [37](see also [26]) showed that geometric weight half-space LPP is a Pfaffian point process (see Section 4.1) and computed the correlation kernel. Later, [21] defined Pfaffian Schur processes generalizing the probability
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